Model for the eruption of the Old Faithful , Yellowstone National Park

Kieran D. O’Hara, Dept. of Earth and Environmental Sciences Several models have been proposed for (Steinberg et al., (emeritus) University of Kentucky, Lexington, Kentucky 40506, 1978; Steinberg, 1980; White et al., 1967; Murty, 1979; Rinehart, USA, [email protected]; and E.K. Esawi, Division of Physical 1980; Ingebritsen and Rojstaczer, 1993, among others) and for the Sciences, Elizabethtown Community & Technical College, Old Faithful geyser in particular (Rinehart, 1965, 1969; Fournier, Elizabethtown, Kentucky 42701, USA 1969; Kieffer, 1984, 1989; Dowden et al., 1991; Kedar et al., 1998). These studies provide substantial insight into the eruption dynamics of geysers in general. The goal of our study is to present ABSTRACT a new physical model for the observed eruption interval of Old A physical model of the Old Faithful geyser successfully repli- Faithful over a continuous period of time for which we have cates the eruption interval for the years 2000–2011. It is based on interval data (both logbook and electronic) as well as duration convective boiling in the conduit in three stages, and the model is and preplay times. Our model differs from previous studies in in good agreement with published time-temperature-depth data. that the heating mechanism of the water occurs by convective The preplay phase, which triggers the main eruption, displays a boiling in three stages; in addition, the length of the preplay Rayleigh probability density function with a mode at nine minutes, phase plays a central role in modeling the geyser eruption interval. and it plays an essential role in determining the main eruption Our study is based on data for the twelve-year period 2000–2011. interval. It is assumed that temperature gradients are small due to Because the data are averaged over time, changes in the eruption convection, and individual convection cells can be assigned a single interval with time due to intermittent or short-term effects heat content. Based on previous observations and drill-hole mea- (e.g., earthquakes or climate effects) are not addressed. The raw surements, the bottom heating and recharge temperatures are data for this study were provided by the Geyser Observation and assumed to be 110 °C and 80 °C, respectively, and the total volume Study Association’s website at http://gosa.org/ofvclogs.aspx of the cylindrical conduit is 23 m3. A prescription for both short (GOSA, 2011). and long eruption intervals is ERUPTION CYCLE eruption interval = (time to boiling of upper stage) + (preplay time). The eruption cycle can be divided into three phases: 1. The eruption duration (2–5 min.); A composite model reproduces the bimodal eruption pattern 2. The eruption interval (or recharge phase, 55–120 min.); in which long eruption durations are followed by long eruption and intervals and short eruption durations are followed by short erup- 3. The preplay phase (small, discrete eruptions lasting tion intervals. The cause of short eruption durations is not ad- 1–35 min.). dressed and remains unresolved. The eruption itself can be subdivided into three stages: INTRODUCTION 1. Initiation and unsteady flow; The Old Faithful geyser, an icon of the American West, has 2. Steady flow; and been studied for more than a century (Hayden, 1872). In 2010, 3. Decline (Kieffer, 1984). 3.6 million tourists visited the Yellowstone National Park, , for a total of 170 million visits since the park was established in The bimodal eruption pattern for the years 2000–2011 is shown 1872, indicating an enduring curiosity by the public in geyser in Figure 1. The eruption interval shows a major mode at ~92 activity. The geyser is not as regular as is commonly thought, minutes and a subsidiary mode at ~60–65 minutes (Fig. 1A). The because the eruption interval is bimodal and has varied over time eruption duration for the same period shows a mode at 4–4.5 (Rinehart, 1980). For example, in 1948 the mean eruption interval minutes with the shorter duration events (2–2.5 minutes) being was 64 minutes (Birch and Kennedy (1972), in 1979 it was relatively rare, accounting for ~5% of all eruptions (Fig. 1B). ~80 minutes (Kieffer, 1984), and in 2011 it was ~92 minutes (GOSA, The eruption interval shows a pattern whereby short eruption 2011). This pattern is attributed to earthquake activity and its effect durations are followed by short recharge intervals, and long erup- on the local hydrology. Lengthening of the eruption cycle and other tion durations are followed by long recharge intervals; this pattern changes in geyser activity have been documented following major has been explained by the geyser’s seismicity (Rinehart, 1965). earthquakes in 1959, 1983, and 2002 (Rinehart, 1972; Hutchinson, Seismic noise produced by Old Faithful is interpreted in terms of 1985; Husen et al., 2004). Decadal and seasonal variations in the the dynamics of vapor bubble formation (and collapse) during y | June 2013 y | June 2013 a eruption interval have also been ascribed to variations in the hydro­ boiling in the conduit (Kieffer, 1984, 1989; Kedar et al., 1996, logical cycle in Yellowstone (Hurwitz et al., 2008). 1998; Cros et al., 2011). That seismic activity resumes almost GSA Tod

GSA Today, v. 23, no. 6, doi: 10.1130/GSATG166A.1. 4 A

Figure 2. Probability histogram for the preplay eruption length for the years B 2000–2011. The data have a mode at 9 min. (standard deviation [s.d.] = 5.6 min.). The distribution can be best modeled by a single parameter Rayleigh probability density function (Eq. 1 in text) with a mode at s = 9 min. (s.d. = 5.2) for x ≥ 0. The anomaly in the data at 1 min. is inferred to be an artifact of observation.

modeled using a single parameter Rayleigh probability density function (x > 0)

(1)

Figure 1. Probability histograms for eruption interval (A) and eruption duration (B) for the years 2000–2011 for Old Faithful. Electronic interval data with a mode (s) at nine minutes and a standard deviation of 5.2 (GOSA, 2011) for this time period agree with the log book data to within minutes (solid curve, Fig. 2). The anomalously large number of 1 min. when weekly averages are compared. one-minute preplay events is attributed to assigning some of the unsteady initiation stage to a short preplay event. In general, Rayleigh distributions have been used to model, among other immediately after short-duration eruptions, but takes substan- phenomena, the height of ocean waves and wind velocities (e.g., tially longer to resume after long eruptions, is consistent with the Abd-Elfattah, 2011). The Rayleigh function fit indicates that the idea that it takes longer for the water to recharge and come to a preplay eruptions are discrete random events. boil after a long eruption (resulting in a seismic quiet period), whereas after a short eruption, the water remaining in the conduit PHYSICAL MODEL is still boiling and emits seismic noise throughout the recharge phase (e.g., Kieffer, 1984). Conduit Geometry A variety of Old Faithful conduit shapes have been proposed to PREPLAY PHASE account for aspects of its eruption characteristics or filling history The preplay phase consists of discrete splashes (each several (Geis, 1968; Rinehart, 1980; Kieffer 1984). Based on down-hole meters in height lasting a few seconds) and precedes the main video camera observations, Hutchinson et al. (1997) indicate that eruption. For example, observations of five consecutive eruptions the immediately accessible conduit is an irregularly shaped enlarged on live video in May 2012 showed the following preplay lengths east-west–trending fracture ~22 m deep. A constriction exists at (in minutes) with the number of discrete splashes in parentheses: ~7 m, above which the water level does not rise, except during 17 (17); 10 (9); 12 (16); 13 (15); and 20 (18), suggesting that preplay preplay and the main eruption itself—presumably because an events occur on average about once a minute. It has long been overflow outlet exists at this depth, possibly connected to the inferred that these discrete events trigger the main eruption by adjacent (Rinehart, 1980). For simplicity, we assume bringing the water in all or part of the conduit onto the boiling here that the conduit is cylindrical with a cross sectional area of curve by reducing hydrostatic pressure in the water column 1.5 m2 for a total volume of ~23 m3 (23,000 liters) (Fig. 3). The (Bunsen, 1845, quoted in Allen and Day, 1935, p. 210), and this conduit recharge phase is divided, for the purposes of calculation idea is still accepted (e.g., Kedar et al., 1998). For the most part, and for comparison with the data of Birch and Kennedy (1972), the discrete nature of these smaller eruptions is easily into three stages (S1, S2, and S3), each with a height of 5 m and a distinguished from the main eruption phase. However, in the case volume of 7.7 m3 (7,700 liters). As discussed later, our model is not of very short preplay times, it becomes a judgment call whether particularly sensitive to the details of the conduit geometry. the unsteady initiation of flow is part of the main eruption itself or represents a short preplay event—this would overestimate the Water Temperatures and Recharge Rate number of short preplay events. The data of Birch and Kennedy (1972) are still the most y | June 2013

The length of the preplay phase was calculated by subtracting important time-temperature-depth information available for Old a the time of the beginning of preplay from the beginning of the Faithful—and their Figure 5 is recast here in Figure 4. This main eruption (Fig. 2). The preplay distribution shows a mode at diagram reveals in-phase heating and cooling episodes at different nine minutes with a standard deviation of 5.6 minutes, and the depths superimposed on an overall heating trend, which we GSA Tod distribution is skewed to the right. This distribution can best be interpret to be due to three distinct convection cells in the conduit 5 Figure 4. Time-temperature plot for a single Old Faithful eruption at four depths (6, 12, 18, and 21 m) based on Figure 5 in Birch and Kennedy (1972). The in-phase cooling and heating at three different depths is interpreted to be due to convective mixing (see Fig. 3). The data indicate that the temperature in individual convection cells is not homogenized over time. Best-fit lines indicate an overall heating trend. Based on these data, a temperature of 110 °C is taken here as the temperature of the heat source at the base of the observable conduit, and the temperature of the recharge water is assumed to be 80 °C. A constant recharge rate of 5 liters per second is also assumed, based on the total volume of discharge and the recharge time (Kieffer, 1984); this rate would fill the conduit up to the 7-m level in ~77 minutes. A constant recharge rate is a reasonable approximation because the hydrostatic pressure in the conduit shows an approximately linear profile over most of the recharge cycle (Hutchinson et al., 1997, their fig. 5A).

Lumped Capacitance Assumption Figure 3. Idealized cylindrical model (radius 0.7 m) for the Old Faithful conduit to a depth of ~22 m with a total volume of 23 m3. The recharge water The process of filling a conduit from above with cold water and temperature is assumed to be 80 °C, and the bottom source temperature is heating it from below at the same time produces an inherently assumed to be 110 °C. Recharge of the conduit occurs in three equal stages: unstable situation. The difference in density of water at 110 °C S , S , and S . Convection cells are based on Figure 4. 1 2 3 compared to 80 °C is ~20 kg/m3, which produces sufficient buoyancy to overcome viscous forces (i.e., high Rayleigh (Fig. 3). The data of Birch and Kennedy (1972, their fig. 5) also numbers), leading to convective overturn (Murty, 1979). The show a maximum temperature of 116 °C at the deepest levels, and temperature variations plotted in Figure 4, and those observed by Rinehart (1969, his fig. 1) observed a maximum temperature of Hutchinson et al. (1997; their fig. 6) are attributed to vigorous ~110 °C during two eruption cycles (but his depth estimate of convective overturn and mixing of parcels of hot and cold water. 30 m may be in error). Over a period of ~30 minutes before an In the context of convection, Newton’s law of cooling relates the eruption, Hutchinson et al. (1997) also observed water heating rate (dQ/dt in W/m2) to the difference in temperature temperatures of ~110 °C near the bottom of the conduit; they between the source temperature and the ambient temperature of interpreted temperature oscillations to be due to convection. the water (DT) at a given pressure: Drilling in the Upper Geyser Basin, to which Old Faithful belongs, showed bottom hole temperatures at a depth of 20 m of 90 °C (2) (Y-7 hole); 135 °C (Y-8); 120 °C (Y-1); and 125 °C (Carnegie I) (White et al., 1975), for a mean temperature of 118 °C at a depth where h is the convective heat transfer coefficient (W/m2K) and equivalent to the base of the Old Faithful conduit. Based on geo­ A is the cross sectional area of the conduit (1.5 m2). chemical arguments and water salinity, Fournier (1979, his fig. 7) If convection dominates over heat conduction and the water indicated the existence of a parent water source of ~200 °C for the within individual convection cells is well mixed so that thermal Upper Basin at depth, but such a high temperature has not been gradients become small, the average thermal energy (Q) of each measured in the accessible conduit of Old Faithful. stage can be specified—it is simply related to temperature (T) by The lowest temperature observed by Hutchinson et al. (1997) the specific heat capacity(C): dQ/dT = C (kJ/kg). This is termed was 86 °C and is interpreted to be due to recharge water the Lumped Capacitance Assumption (Incropera and DeWitt, y | June 2013 y | June 2013

a percolating into the conduit from shallower levels. Drilling in the 2005, p. 240), and because of the vigorous convective mixing Upper Geyser basin also indicates bottom hole rock temperatures observed in the conduit, this approximation appears to be at a depth of 7 m in the range of 60–100 °C (White et al., 1975). justified. The change in heat energy with time is then dQ/dt = GSA Tod The lowest temperature reported by both Birch and Kennedy CdT/dt. Substituting this relationship into equation (2) gives the (1972, their fig. 5) and Rinehart (1969, his fig. 1) was 93 °C. change in temperature with time: 6 (3) and the solution to equation (3) is

(4) where DTo is the temperature difference at time t = 0 and the constant k = hA/mC, where m is the mass of the water. It is assumed that at the end of each recharge stage the reservoirs are well mixed due to convection and each stage can be assigned a 110 °C. Stage 2 is heated from below at 105 °C, which is the mean single heat content and a temperature. This allows the thermal temperature of S1 during S2 filling. Stage 3 is heated from below at evolution of each stage to be modeled individually using the 99 °C, which is the mean temperature of S2 during S3 filling. Lumped Capacitance Assumption. Two methods of heating were used—one heated the entire volume of water at each stage (7.7 m3) as a function of time (Fig. 5, Nucleate Boiling solid curves), and the second method heated the water while incrementally increasing the volume by 5 liters per second (Fig. 5, During convection, rising parcels of hot water will dashed color curves). The incremental heating produces an initial spontaneously boil at lower water pressures. Vapor bubbles rapid temperature rise in the early part of the heating history, but nucleate and rise, causing liquid mixing, effectively transporting after ~25 minutes, the two curves converge for each stage. Also heat upward. On reaching the cooler water, the rising bubbles shown on Figure 5 are the best fit lines from Figure 4 at different collapse and give up their heat to the surrounding water depths, based on the data of Birch and Kennedy (1972). Both (collapsing bubbles also then produce the seismic noise referred to heating methods produce a good fit to the data in the later part of earlier). These processes are referred to as nucleate boiling, and it the heating history. This allows a constant value for the mass of is a very effective heat transfer mechanism (Incropera and DeWitt, water (m) to be used in equation (4) (Fig. 5, solid curves). From a 2005, p. 599). thermodynamic point of view, because enthalpy is a state function Because of the complexity of analytical solutions during (Smith, 2005), the final temperature of each stage is independent nucleate boiling, values for the convective heat transfer coefficient of the heating path so that both heating methods produce the h are usually derived empirically. Experimental evidence indicates same final result. that h increases as DT in equation (2) increases, where DT is now The temperature of the water at each stage as a function of the difference between the boiling temperature and the heating depth and time can be evaluated from Figure 5. Figure 6 shows source temperature. The experiments indicate that nucleate the temperature of S , S , and S at 75 minutes and at 85 minutes boiling occurs when DT is 5 °C to 30 °C and produces values of 1 2 3 into the recharge phase. Also shown in Figure 6 is the reference dQ/dt in the range of 104–106 (W/m2); when DT = 10 °C, dQ/dt ≈ 105 boiling point curve (solid heavy line) and the depth- (Incropera and DeWitt, 2005, p. 598). temperature curve of Birch and Kennedy (1972) 2.5 minutes In the present case, nucleate boiling would occur if a parcel of before eruption (dashed heavy line). After 75 minutes into the water at 11 m depth (boiling point of 105 °C) rose to a depth of recharge phase, all three stages are below the boiling curve, but ~7.5 m in the conduit (boiling point of 95 °C). Direct evidence in at 85 minutes the uppermost stage (S ) has reached boiling. support of this process comes from the existence of superheated 3 geyser waters throughout Yellowstone National Park (Allen and Day, 1935, their table 3; Bloss and Barth, 1949) and Old Faithful itself (Hutchinson et al., 1997). A preliminary estimate for the value of h can be made by taking DT in equation (2) as 10 °C, based on the example above, which corresponds to a value of dQ/dt of 105 (W/m2) based on experimental data. This produces a value for h of 6.6 × 103 W/m2K. A value approximately twice this amount, 1.2 × 104 W/m2K, produces the best fit to the data of Birch and Kennedy (1972) in the following models, and this value is used here (Table 1). The sensitivity of our model to this value is discussed later.

MODEL RESULTS Figure 5 shows the temperature versus time evolution for each Figure 5. Temperature evolution with time (solid curves) for the three stages, based on equation (4) in the text using the parameters in Table 1. Stage 1 begins of the three stages using equation (4). Stage one (S1) begins filling

filling at t = 0, followed by stage 2 at ~25 min. and stage 3 at ~50 min. The y | June 2013

at t = 0 at a rate of 5 liters per second, and this takes ~25 minutes, a colored dashed curves indicate the heating path when the conduit is at which time S begins to fill. This takes an additional 25 minutes, 2 incrementally filled at a rate of 5 liters/s, using equation (4). After ~25 min. after which time S3 begins to fill; all three stages are full after into each stage, both solid and dashed curves converge and become identical. GSA Tod ~77 minutes. The temperature at each stage increases with time The best fit lines from Figure 4 at different depths are indicated—the heating from its initial value of 80 °C. Stage 1 is heated from below at curves are a good fit to the data. 7 Figure 7. Temperature-time evolution of the upper stage (S3) and middle stage

(S2) after a short eruption (upper axis) and the temperature-time evolution of

the lower stage (S1) after a long eruption (lower axis). The upper stage takes only 55 min. to reach boiling, which begins the preplay stage. Preplay activity triggers a full eruption, as described in the text.

below the surface, this continuous boiling will not produce any Figure 6. Temperature in the conduit after 75 and 85 min. (vertical dashed preplay activity. The continuous boiling accounts for the contin­ lines), based on Figure 5, relative to the boiling curve. The water boils at 93 °C uous seismic noise after a short eruption but absent after a long

at the park elevation (2246 m). The uppermost stage (S3) intersects the boiling eruption (Kedar et al., 1998). After a short eruption, the second curve after 85 min. The heavy dashed curve represents the data of Birch and stage will begin to refill and heat as before, followed 25 minutes Kennedy (1972), 2.5 min. before eruption. The temperature in each stage after later by the uppermost stage. 85 min. is consistent with these data. Figure 7 shows the temperature evolution of all three stages after a short eruption using the same parameters as before (Table 1). The middle and lower stages are still below the boiling curve at The upper axis refers to the time after a short eruption for S2 and 85 minutes, consistent with the Birch and Kennedy (1972) data S3, whereas the lower axis refers to the time after a long eruption 2.5 minutes before the eruption. for S1. The upper stage (S3) reaches boiling (~93 °C at this elevation) within 55 minutes. This boiling initiates the preplay Long and Short Eruption Intervals phase, causing S2 to boil as hydrostatic pressure is released on the entire water column. The recipe for a short eruption interval is the Boiling at the top of the upper stage (S3) will cause small amounts of water to be emitted at ground level, thereby reducing same as before but with a shorter heating time for S3 (compare S3 the pressure on deeper water, which in turn causes more boiling at in Figs. 5 and 7). deeper levels (Kieffer, 1989)—this is the beginning of the preplay In the case of the long eruptions, the Rayleigh preplay phase. The net result of these small emissions is that the pressure distribution (mode = 9 min.) is superimposed on an 85-minute reduction propagates downward until all the water column is at or heating phase and a 55-minute heating phase for short eruptions, above boiling—this triggers the main eruption phase, which producing the composite distribution shown in Figure 8. The occurs according to the following recipe: agreement with the observed data (Fig. 1A) is quite good, with a subsidiary mode at ~60 minutes for short eruption intervals and a Eruption interval = refill and boiling time for mode at ~92 minutes for long eruption intervals. The physical cause of short-duration eruptions remains unresolved. S3 + preplay time (Rayleigh distribution). DISCUSSION Refill and boiling time after a long eruption for S3 is typically 85 minutes, based on our model (Fig. 5). The pressure data of An alternative model with a conduit geometry based on that Hutchinson et al. (1997) shows a pressure drop of 0.2 bars illustrated by Hutchinson et al. (1997) yields very similar results to (20 kPa) immediately before an eruption, corresponding to the those presented here, indicating that the model is not particularly emission of 2 m of water (or about 3000 liters). This pressure sensitive to the conduit geometry. On the other hand, the model is drop is interpreted to be part of the preplay phase that triggered quite sensitive to parameters such as the absolute value of the the main eruption. source temperature and the recharge temperature—changes as Short eruption intervals follow short eruptions. This is usually small as ±20% in these values produce a substantially worse fit to rationalized on the basis that short eruptions expel smaller the data of Birch and Kennedy (1972); these temperatures, amounts of water and therefore the conduit takes a shorter time however, are reasonably well constrained. to recharge and heat to boiling before the next eruption. For illus­ Changes in the value of the convective heat transfer coefficient y | June 2013 y | June 2013

a tration purposes (Fig. 7), it is assumed here that short eruptions (h) of ±20% also produce substantially worse fits to the data. The

involve only the second and third stages (S2 and S3), but any large value we chose (based on DT = 10 °C) is slightly above the middle fraction of the total volume could be expelled. If a short eruption of the experimental range for the nucleate boiling regime GSA Tod expels these two stages, the remaining contents (namely, S1) (Incropera and DeWitt, 2005, p. 598). Lastly, Hutchinson et al. continue to heat toward boiling. Because the water level is well (1997) suggested that CO2 may be present in the conduit water 8 Hayden, F.V., 1872, Preliminary report of the United States Geological Survey of Montana and portions of adjacent Territories; being a fifth annual report of progress: Part I: Washington, D.C., U.S. Government Printing Office, p. 13–204. Hurwitz, S., Kumar, A., Taylor, R., and Heasler, H., 2008, Climate-induced variations of geyser periodicity in Yellowstone National Park, USA: Geology, v. 36, p. 451–454, doi: 10.1130/G24723A.1. Husen, S., Taylor, R., Smith, R.B., and Healser, H., 2004, Changes in geyser eruption behavior and remotely triggered seismicity in Yellowstone National Park produced by the 2002 M7.9 Denali fault earthquake, Alaska: Geology, v. 32, p. 537–540, doi: 10.1130/G20381.1. Hutchinson, R.A., 1985, Hydrothermal changes in the Upper Geyser Basin, Figure 8. By adding 55 min. (short eruption heating time) and ~85 min. (long Yellowstone National Park, after the 1983 Borah Peak, , earthquake, eruption heating time) to the preplay Rayleigh distribution (Eq. [1], with a in Stein, R.S., and Bucknam, R.C., eds., Proceedings of Workshop 28 on mode of 9 min.; solid curves), the distribution of short and long eruption the 1983 Borah Peak, Idaho, Earthquake: U.S. Geological Survey Open- intervals are reproduced quite well for the period 2000–2011. The circular File Report 85-290-A, p. 612–624. symbols are the same data as in Figure 1A. Because short eruptions occur about Hutchinson, R.A., Westphal, J.A., and Kieffer, S.W., 1997, In situ observations of 5% of the time, the Rayleigh probability was scaled by a factor of 0.05 relative Old Faithful Geyser: Geology, v. 25, p. 875–878, doi: 10.1130/0091-7613 to long eruption intervals. (1997)025<0875:ISOOOF>2.3.CO;2. Incropera, F.P., and DeWitt, D.P., 2005, Fundamentals of Heat and Mass Transfer, 5th edition: New York, Wiley, 1008 p. with a pressure of up to 0.2 bars (20 kPa), which would act to Ingebritsen, S.E., and Rojstaczer, S.A., 1993, Controls on geyser periodicity: lower the reference boiling curve. On Figure 6, this corresponds to Science, v. 262, p. 889–892, doi: 10.1126/science.262.5135.889. ~2 m of hydrostatic pressure and would bring the reference curve Kedar, S., Sturtevant, B., and Kanamori, H., 1996, The origin of harmonic tremor at closer to the P-T conditions in the conduit by this amount Old Faithful geyser: Nature, v. 379, p. 708–711, doi: 10.1038/379708a0. (equivalent to 3000 liters of water). The overall effect would be Kedar, S., Kanamori, H., and Sturtevant, B., 1998, Bubble collapse as the source equivalent to shortening the preplay phase (and the eruption of tremor at Old Faithful Geyser: Journal of Geophysical Research, v. 103, p. 24,283–24,299, doi: 10.1029/98JB01824. interval) by a few minutes. 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